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Optical simulations within and beyond the paraxial limit 1 Daniel Brown, Charlotte Bond and Andreas Freise University of Birmingham

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We need to know how to accurately calculate how distortions of optical elements effect the beam Simulating realistic optics 2 Surface and bulk distortions Finite element sizes Thermal effects Manufacturing errors Mirror maps Gaussian beam Higher order modes like Laguerre- Gaussian (LG) beams Beam clipping Ideal beam Final distorted beam

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Methods to simulate light Fast-Fourier Transform (FFT) methods Solving scalar wave diffraction integrals with FFTs Modal method Represent beam in some basis set, usually eigenfunctions of the system Rigorous Simulations What to use when the above breakdown? Resort to solving Maxwell equations properly 3

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FFT Methods input fieldmirror surfaceoutput field The effect of a mirror surface is computed by multiplying a grid of complex numbers describing the input field by a grid describing a function of the mirror surface. An FFT method commonly refers to solving the scalar diffraction integrals using Fast Fourier Transforms Can quickly propagate beam through complicated distortions, useful when studying non- eigenmode problems Quasi-time domain, cavity simulations can require computing multiple round trips to find steady state

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Diffraction mathematically based on Greens theorem, then making many approximations to make it solvable Scalar Diffraction 5 [1] Introduction to Fourier Optics, Goodman [2] Shen 2006, Fast-Fourier-transform based numerical integration method for the Rayleigh–Sommerfeld diffraction formula [3] Nascov 2009, Fast computation algorithm for the Rayleigh–Sommerfeld diffraction formula using a type of scaled convolution Helmholtz-Kirchhoff integral equation Rayleigh-Sommerfeld (RS) integral equation Fresnel Diffraction Fraunhofer Diffraction Number of approximations Solve with FFTs [1][2][3] Solve with Numerical Integration Paraxial Approximations

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Scalar Diffraction 6 [1] Harvey 2006, Non-paraxial scalar treatment of sinusoidal phase gratings Smooth surface, small h/d Rough surface, large h/d

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Fresnel and Fraunhofer conditions 7 Fresnel Diffraction Fraunhofer Diffraction Conditions on distance from aperture [1] Conditions on distance from beam axis If you are too close to the aperture or looking at a point too far from the beam axis, you should be using Rayleigh-Sommerfeld Diffraction! [1] Angular criterion to distinguish between Fraunhofer and Fresnel diffraction, Medina 2004

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Testing difference between Rayleigh-Sommerfeld and Fresnel diffraction Small cavity Small mirrors and short Larger angle involved LIGO arm cavity Big mirrors and long Small angles involved Khalili Cavity Big mirrors and short Larger angles involved Examples 8 L=0.1m L=4km L=3m 34cm 2.5cm

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How do the conditions look for each example Examples 9 L=0.1m L=4km L=3m 34cm 2.5cm Max angle

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Round trip beam power difference between Rayleigh-Sommerfeld and Fresnel FFT Examples 10 L=0.1m L=4km L=3m 34cm 2.5cm Cavity field of view Power difference Largest power difference only seen when beam size is very large!

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But other things do not work 11

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FFT Aliasing with tilted surfaces 12

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Modal models n m Modal model only deals with paraxial beams and small distortions, what we would expect in our optical systems 13

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Why we use modal models 14 Model arbitrary optical setups Can tune essentially every parameter Quick prototyping Fast computation for exploring parameter space Undergone a lot of debugging and validation Finesse does all the hard work for you! Currently under active development Mirror map or some distortion

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Sinusoidal grating with modes 15 [1] Winkler 94, Light scattering described in the mode picture

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Sinusoidal grating with modes 16 Low spatial frequency compared to beam size High spatial frequency compared to beam size For low spatial frequency distortions only a few modes are needed As spatial frequency becomes higher, higher diffraction orders appear which modal model can’t handle. Only 0 th order is accurately modeled Modes struggle with complicated beam distortions, requires many modes 80ppm is the power in the 1 st orders

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Real life modal model examples A lot of work done on many topics: Thermal distortions Mirror maps Cavity scans Triangular mode cleaners Simulating LG33 beam in a cavity with ETM mirror map Modal simulation done with only 12 modes hence missing 13 th mode peak compared to FFT 17

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Rigorous Simulations Scalar diffraction and modal methods can’t do everything, so what next? Wanted to study waveguide coatings in more detail Sub-wavelength structures Polarisation dependant Electric and magnetic field coupling Not paraxial Requires solving Maxwell equations properly for beam propagation Method of choice: Finite-Difference Time-Domain (FDTD) 18

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Waveguide Coating – Phase noise Waveguide coatings proposed for reducing thermal noise Gratings however are not ideal due to grating motion coupling into beams phase [1] Wanted to verify waveguide coatings immunity [2] to this grating displacement phase variations with finite beams and higher order modes 19 [1] Phase and alignment noise in grating interferometers, Freise et al 2007 [2] Invariance of waveguide grating mirrors to lateral beam displacement, Freidrich et al 2010

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Finite-Difference Time-Domain (FDTD) The Yee FDTD Algorithm (1966), solving Maxwell equations in 3D volume Less approximations made compared to FFT and modal model Calculates near-field very accurately only, propagate with scalar diffraction again once scalar approximation is valid Taflove, Allen and Hagness, Susan C. Computational Electrodynamics: The Finite-Difference. Time-Domain Method, Third Edition 20 The idea is to… 1.Discretise space and insert different materials 2.Position E and B fields on face and edges of cubes 3.Inject source signal 4.Compute E and B fields using update equations in a leapfrog fashion. Loop for as long as needed

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Waveguide coating simulation Simulation programmed myself, open to anyone else interested in it Validate simulation was working, compare to Jena work Analyse reflected beam phase front with varying grating displacements to look for phase noise 21 Power flow measured across this boundary Incident field injected along TFSF boundary Only beam reflected from grating here

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Waveguide Coating Simulations 23 TE ReflectivityTM Reflectivity s – waveguide depth, g – grating depth [1] High reflectivity grating waveguide coatings for 1064 nm, Bunkowski 2006

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Conclusion… FFT’s work, just need to be careful, consider using RS in certain cases Modal model works, just need to ensure you use enough modes Modal model and FFT are identical for all real world examples Seen one option for rigorously simulating complex structures using the FDTD FDTD code is available to anyone who wants to use/play with it! 24

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…and finished… 25

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